School of Bwm.edi.cine, Far Easter Federal University 690950, Vladivostok, Russia

Received November 17, 2014

The Einstein-de Haas effect reveals a transfer of anguar momentum from microscopic constituents (electrons) to a macroscopic body, but in the case of massless fermions, one could expect the transfer of the chirality of constituents to macroscopic helical motion. For such a picture to be consistent, the macroscopic helicity is to be conserved classically, to echo the conservation of the angular momentum of a rotating body. The helicity conservation would in turn impose constraints on hydrodynamics of chiral liquids (whose constituents are massless fermions). Essentially, the chiral liquids are dissipation-free, on the classical level. Reservations and alternatives to this scenario are discussed.

Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday

DOI: 10.7868/S0044451015030106

1. INTRODUCTION. CHIRAL LIQUIDS

Theory of liquids with massless fermionic constituents has been greatly highlighted recently (for a review, see, e. g., lecture volume [1]). The interest in such chiral liquids was triggered by the discovery of QCD plasma, with its nearly massless quarks (see, e.g., [2]). The quark gluon plasma exhibits remarkable properties. In particular, it is characterized by a low ratio of the viscosity to the entropy density ,s\ close to its conjectured quantum lower bound [3]. However, this property of the quark gluon plasma has not yet been related to the (nearly) chiral nature of the plasma, and we return to this point later.

Vector and axial-vector currents are natural probes of the chiral nature of the underlying field theory. Moreover, from the theoretical standpoint, the consideration of chiral media with an asymmetric right left composition or a nonvanishing chiral chemical potential //,5 0 represents an especially clean case. In particular, one predicts the existence of the chiral magnetic

E-mail: vzakharov&itep.ru

effect [4 6], or a flow of electric current along the magnetic field in equilibrium,

j^crMB,, (1)

where Dfl = (l/2)eilpQ^«"FQ^, u'1 is the 4-velocity of an element of the liquid, and FQ/ii is the standard electromagnetic field tensor. In the rest frame, B^ reduces to the magnetic field. We mostly focus on the vortical chiral effect [7 9], according to which helical macroscopic motion of the liquid contributes to the axial current -fi-i1

.ifi = {ll^Oue^pvUvdpUv. (2)

We note that is actually a function of both the chemical potential and temperature. For simplicity, we mostly suppress the temperature dependences. This does not affect our conclusions.

Currents (1) and (2) are predicted to exhibit remarkable properties. First, the coefficients <rM and au are uniquely determined in terms of the chiral anomaly. Thus, for a single massless Dirac fermion with an electric charge e,

where //.5 = //.£ — fin is the chiral chemical potential. For the vortical conductivity au, we obtain

where //. = /¡l + fin- Amusingly, Eqs. (1) and (3) imply that the laws of classical electrodynamics are modified for chiral liquids.

Another intriguing feature of chiral liquids is that currents (1) and (2) are nondissipative. This conclusion already follows from the observation that the currents exist in equilibrium. Another way of reasoning [10] is that both the r.h.s. and the l.h.s. of (1) are odd under time reversal. This is a strong indication that the dynamics behind Eqs. (1) and (2) is Hamiltonian and there is no dissipation. For a discussion of the analogy with supeconductivity, we refer the reader to [11].

As mentioned above, the numerical values of <rM and au can be traced back to the coefficients in front of the product of electric and magnetic fields in the expression for the famous chiral anomaly ("12]:

d>4 = (5>

where the definition of the magnetic field adjusted to the consideration of hydrodynamics is given above, while the electric field in the medium is defined as Ea = u^Fffa- In the hydrodynamic approximation, relations (3) and (4) were originally obtained in Ref. [9]. In their approach, the authors of [9] start with both electric and magnetic external fields present and then let Ea —¥ 0. Remarkably enough, currents (1) and (2) survive in the limit of chiral anomaly (5) being switched off by taking the limit Ea —¥ 0. This implies that already in the limit of the electromagnetic coupling aci —¥ 0 the conserved axial charge is modified in the hydrodynamic setup.

The reason for such a modification can be explained in a number of ways (see in particular, [13 17]). What is specific for hydrodynamics, is the change of the original Hamiltonian H0 of the system to a modified one:

Ho^Ho^ fiQ, (6)

where //. is the chemical potential associated with a conserved charge Q. As a result, there is a change already in the conserved axial current (i. e., in the limit of vanishing electromagnetic coupling). In a somewhat simplified form, the axial charge within the hydrodynamic approach is given by

Q'hydro = Q'naivc + 'Hfluid + 0(e), (7)

where Q',laivc counts the number of elementary chiral constitutents and the fluid helicity is Hfiuid = = f d?'x fi2uio, where uia = (1/2)eaij1su':id1us and we reserve for the possibility of the chemical potential varying in space1 K

The conservation of hydrodynamic axial charge (7) suggests a possibility of transition of the chirality of the constituents into helical macroscopic motion of the liquid. As is mentioned in the abstract, this is an analog of the Einstein de Haas effect. A new point is what can be called the clash of symmetries: 011 the microscopic level, chirality is conserved, but 011 the macroscopic level, we are using the standard hydrodynamic description, which does not incorporate the conservation of chirality in general and was originally developed for nonrelativistic motion of the constituents.

One way to resolve this contradiction is to impose extra constraints 011 the hydrodynamic description [18]. Generically, the solution of these constraints is that classically chiral liquids are dissipation-free. In particular,

''¡classical = 0. (8)

We note that phenomenological consequences from the (hypothesized) conservation of fluid helicity were studied in great detail in magnetohydrodynamics2^ (see, e.g., [19] and the references therein).

The outline of this paper is as follows. In Sec. 2, we discuss the issue of the conservation of macroscopic helical motion in hydrodynamics in more detail. The main conclusion is that the conservation of the axial charge implies dissipation-free hydrodynamics of chiral liquids in the classical approximation. In Sec. 3, we discuss reservations and problems.

2. AXIAL CHARGE IN HYDRODYNAMICS

2.1. Hydrodynamics as an effective field theory

Hydrodynamics is a unversal framework to describe motions in the infrared limit, when the wave lengths of perturbations are much larger than the mean free path of constituents. The beauty of this approach is that hydrodynamic equations of motion reduce to ge-

11 For simplicity, we quote the expression for the fluid helicity in flat space. In curved space, there is an extra geometric factor of y/—g in the integrand.

2) Note, however, that in magnetohydrodynamics, the electromagnetic field is considered to be dynamical, while many results we are quoting refer to the case of global symmetries, or external magnetic and electric fields.

noral conservation laws. In particular, in the absence of external fields, these equations are

= 0, = 0,

а))

where Ttll, is the energy momentum tensor and is a set of conserved currents3).

Since explicit expressions for Tfll, and jjP involve phenomenological expansions in derivatives, hydrodynamics is usually considered as a "typical" effective field theory. However, apart from integrating out hard, or ultraviolet degrees of freedom, the hydrodynamic approximation also assumes a change of language. Indeed, the fi.Q term in hydrodynamic Hamiltonian (6) does not correspond literally to any integration over fundamental interactions and the very notion of the chemical potantial can be introduced only on average, or thcrmodynamically (see, e.g., [20]). Also, the problem we are considering here is somewhat specific since we need a closed expression for the axial charge, with no further contributions [13] from the gradient expansion.

The simplest way to argue that the hydrodynamic axial charge contains extra pieces, see (7), is as follows [13]. We first assume the chemical potential to be small, such that the fi.Q term in hydrodynamic Hamiltonian (6) can be treated as a perturbation. Using the relation SL = SH for a small variation of the La-grangian, we find for small //:

(SL) hydro = ¡ша]а, (9)

where the charge Q above is related to the current ja hi the standard way, Q = f d?'xj0■ Finally, using the analogy with the electromagnetic interaction, 6Lci = в f d4x Aaja, we come to the substitution

In more detail, we recall that chiral anomaly (5) can be reformuated [21] as the statement that the actually-conserved axial charge contains a term with external elect romagnet ic p ot cnt ials:

l l Q conserved = Q-naive + "TZ^Hmagni ^^

For a moment, we ignore possible quantum anomalies. Moreover, we consider only l"(l) anomalies, and then we can redefine the anomaly as a new conserved charge, such that the external electromagnetic field has a non-vanishing axial charge if E'Y Bn # 0.

where we introduce the notation H,, „,.,,,. common in papers on magnetohydrodynamics, which stands for the magnetic helicity,

^~L magn — j $ .1 € ^ A^Fj^ ,

where (i.j, k) range over 1, 2, 3, A; and Fjt are the electromagnetic potential and field